Theorem pm5.6r 939

Description: Conjunction in antecedent versus disjunction in consequent. One direction of Theorem *5.6 of [WhiteheadRussell] p. 125. If ps is decidable, the converse also holds (see pm5.6dc 938). (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
pm5.6r	|- ( ( ph -> ( ps \/ ch ) ) -> ( ( ph /\ -. ps ) -> ch ) )

Proof of Theorem pm5.6r

Step	Hyp	Ref	Expression
1		pm2.53 734	. . 3 |- ( ( ps \/ ch ) -> ( -. ps -> ch ) )
2	1	imim2i 12	. 2 |- ( ( ph -> ( ps \/ ch ) ) -> ( ph -> ( -. ps -> ch ) ) )
3	2	impd 254	1 |- ( ( ph -> ( ps \/ ch ) ) -> ( ( ph /\ -. ps ) -> ch ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117

This theorem is used by:  ssundifim  3611  ballotfilemfc0  13232  ballotfilemfcc  13233